OrthogonalMorphisms 2008-10-14 20:41:03 2008-10-16 16:33:02 Definition orthogonal \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} % define commands here \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} A morphism $f:A\to B$ in a category $\mathcal{C}$ is said to be \emph{orthogonal} to a morphism $g:C\to D$ in $\mathcal{C}$, written $$f\perp g$$ if whenever we have a commutative diagram $$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \\ C \ar[r]_g & D}$$ there is a unique morphism $h:B\to C$ such that the diagram $$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \ar@{.>}[dl]|h \\ C \ar[r]_g & D}$$ is commutative also. If $f\perp g$, we sometimes call the ordered pair $(f,g)$ a \emph{diagonally polar pair}. For example, in \textbf{Set}, the category of sets, any surjective function is orthogonal to an injective function. To see this, suppose $f:A\to B$ is surjective and $g:C\to D$ injective, with $y\circ f= g\circ x$, where $x:A\to C$ and $y:B\to D$ are functions. For any $b\in B$, there is some $a\in A$ such that $f(a)=b$ since $f$ is surjective. Define $h:B\to C$ by $h(b)=x(a)$. Now, if there is $c\in A$ such that $b=f(a)=f(c)$, then $g(x(a))= y(f(a))=y(b)=y(f(c))=g(x(c))$. Since $g$ is injective, $x(a)=x(c)$. This shows that $h$ is a well-defined function. It is clear that $h\circ f=x$ and $g\circ h=y$. Now, if $e:B\to C$ is another such a function, then $g(e(b))=y(b)=g(h(b))$, so that $e(b)=h(b)$ since $g$ is injective. This shows that $h$ is uniquely defined. Here are some basic properties of the orthogonality relation on morphisms: \begin{itemize} \item If either $f$ or $g$ is an isomorphism, then $f\perp g$. \item If $f\perp f$, then $f$ is an isomorphism. \item If $f\perp g$ and $f\perp h$, then $f\perp (h\circ g)$. Similarly, $g\perp f$ and $h\perp f$ imply $(h\circ g)\perp f$. Of course, both statements make sense provided that $h\circ g$ exists. \end{itemize} More generally, if $\mathcal{F}$ and $\mathcal{G}$ are two classes of morphisms in a category $\mathcal{C}$, we say that $\mathcal{F}$ is \emph{orthogonal} to $\mathcal{G}$, or that $(\mathcal{F},\mathcal{G})$ is a \emph{diagonally polar pair}, written $\mathcal{F}\perp \mathcal{G}$, if $f\perp g$ for every $f$ in $\mathcal{F}$ and every $g$ in $\mathcal{G}$. For every class $\mathcal{X}$ of morphism, the largest class of morphisms in $\mathcal{C}$ such that $\mathcal{X}$ is orthogonal to is denoted by $\mathcal{X}_*$, and the largest class of morphisms that is orthogonal to $\mathcal{X}$ is denoted by $\mathcal{X}^*$. Based on the properties of $\perp$ above, below are some properties of $^*$ and $_*$: \begin{itemize} \item $\mathcal{X} \subseteq \mathcal{Y}_*$ iff $\mathcal{Y}\subseteq \mathcal{X}^*$. Equivalently, if $\mathscr{M}$ is the class of all subclasses of morphisms of $\mathcal{C}$, then $(-^*,-_*)$ is a Galois connection between $(\mathscr{M},\subseteq)$ and $(\mathscr{M},\supseteq)$. \item A morphism is in both $\mathcal{X}^*$ and $\mathcal{X}_*$ iff it is an isomorphism. \item Both $\mathcal{X}^*$ and $\mathcal{X}_*$ are closed under $\circ$. \item Given that $m=m_1\circ m_2$ exists in $\mathcal{C}$ and $m_2\in \mathcal{X}_*$, then $m \in \mathcal{X}_*$ iff $m_1\in \mathcal{X}_*$. \item If $f\in \mathcal{X}_*$, then the pullback of $f$ along any morphism is again in $\mathcal{X}_*$. \end{itemize} \begin{thebibliography}{9} \bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994) \end{thebibliography}